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Theorem bnj96 28408
Description: Technical lemma for bnj150 28419. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj96.1  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
Assertion
Ref Expression
bnj96  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
Distinct variable groups:    x, A    x, R
Allowed substitution hint:    F( x)

Proof of Theorem bnj96
StepHypRef Expression
1 bnj93 28406 . . 3  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  pred ( x ,  A ,  R )  e.  _V )
2 dmsnopg 5181 . . 3  |-  (  pred ( x ,  A ,  R )  e.  _V  ->  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }  =  { (/) } )
31, 2syl 15 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }  =  { (/) } )
4 bnj96.1 . . 3  |-  F  =  { <. (/) ,  pred (
x ,  A ,  R ) >. }
54dmeqi 4917 . 2  |-  dom  F  =  dom  { <. (/) ,  pred ( x ,  A ,  R ) >. }
6 df1o2 6533 . 2  |-  1o  =  { (/) }
73, 5, 63eqtr4g 2373 1  |-  ( ( R  FrSe  A  /\  x  e.  A )  ->  dom  F  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822   (/)c0 3489   {csn 3674   <.cop 3677   dom cdm 4726   1oc1o 6514    predc-bnj14 28224    FrSe w-bnj15 28228
This theorem is referenced by:  bnj150  28419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-br 4061  df-suc 4435  df-dm 4736  df-1o 6521  df-bnj13 28227  df-bnj15 28229
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